INSTITUTE FOR FUSION STUDIES

DE-FG03-96ER-54346-782 IFSR #782

A Photon Accelerator-Large Blueshifting of Femtosecond Pulses in Semiconductors

V.I. BEREZHIANI International Centre for Theoretical Physics, 34100 Trieste, Italy

S.M. MAHAJAN Institute for Fusion Studies

and I.G. MURUSIDZE

Institute of Physics, 380077, Tbilisi, Georgia

The University of Texas at Austin, Austin, TX 78712

April 1997 I

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A Photon Accelerator-Large Blueshifting of Femtosecond

Pulses in Semiconductors

V.I. Berezhiani

International Centre for Theoretical Physics, 34100 Trieste, Italy

S.M. Mahajan

Institute for Fusion Studies: The University of Texas at Austin, Tx 78712

I.G. llurusidze

Institute of Physics, 380077 Tbilisi, Georgia

(March 18, 1997)

Abstract

The availability of relatively high intensity ( I > 109Wcm-*) [but moderate

’ - nJ) total energy], femtosecond laser pulses with wavelengths ranging from

the ultraviolet to the rnid-infrared [l] has opened the doors for a serious in-

vestigation of the nonlinear optical properties of matter on ultrashort time

scales in a new parameter regime. Even small intensity-dependent noniinear-

ities can begin to play a major role in the overall electrodynamics, and in

determining the fate of the propagating pulse. It is shown that a femtosecond

pulse propagating near a two-photon transition in a semiconductor waveguide

can undergo a large blueshift.

1

In this paper we study the propagation of j, short femtosecond laser pulse in a semi-

conductor when two-photon absorption (TPA) is dominant. TPA is a nonlinear resonant

process in which an interband transition is induced by the simultaneous absorption of a pair

of photons. For photon energies in excess of half the band gap energy, TPA generates free

carriers which contribute to nonlinear refraction [2]. In conventional approaches, the TPA

contribution is limited to the imaginary part of the third-order susceptibility $3) which

determines the third-order induced polarization P(3) in the wave equation [3]

.

where k = nod/c, n o is the packground refractive index, ug is the pulse group velocity in

the medium, and E is the slowly varying envelope of the pulse electric field. Equation (I),

derived by invoking the slowly-varying-envelope approximation (SVEA) , is an adequate de-

scription only of those processes which happen at time scales much greater than the wave

period. For instance, the nonlinear refractive and absorption properties described by P(3) .

and the effects induced by them, such as self-phase modulation (SPM) and nonlinear damp-

ing due to TPA, will manifest themselves after the pulse has travelled several pulse lengths

through the medium. Equation (1) also neglects the groupvelocity dispersion rVD). These

considerations limit the applicability of Eq. (1) to situations for which the TPA generated

excess carrier plasma density is far below the critical value at which the laser frequency is

equal to the carrier plasma frequency.

New models, designed to simulate physics on time scales comparable to or shorter than

the inverse of the pulse width, axe thus needed for an adequate description of the fast interac-

i ion between semiconductors and high intensity femtosecond laser pulses. An understanding

of Fhese processes will help in the exploitation of the large and extremely fast optical non-

linearities in semiconductors which axe .;o attractive for optical devices. The TPA model

developed in this paper ago reveals se. .-a1 interesting and specific aspects of this interac-

tion which are not found in the vast extant literature on the nonlinear optical properties of

semiconductors [3,4].

2

Concentrating on TPA, we pick the incident laser photon energy tiW to lie in the range

E,/2 < fw < Eg, where Eg is the band-gap energy. The laser pulse is tuned near the

two-photon resonance well below the lowest one-photon transition threshold. Propagating

through the semiconductor, the pulse, via the two-photon absorption, places a large number

of mobile electrons in the conduction band. The density N of this newly created plasma is

determined by

.

where I is the laser intensity inside the semiconductor, cr = 32/2fiw, and 3 2 is the two-

photon absorption coefficient. The laser pulse duration TL is assumed to be much less than

the characteristic response time of the medium. Effects of recombination and diffusion are,

therefore, neglected. It should be noted that the carrier recombination (both radiative and

nonradiative) takes place typically on nanosecond or even longer time scales [2]. These times

are much longer than the time scales of the duration of the processes described by our model.

However radiative recombination may be enhanced in the presence of an intense radiation

field. This complicated effect is Ieft for a future investigation.

Traditionally. TPA is taken to be a source for generating free charge carriers, even a

large number of them. The energy expended in this process results in an attenuation of the

laser pulse. In our model, however, we stress a complementary aspect of the interaction,

the acceleration of the newly created free carriers and the resulting currents ( fa t ly varying)

induced in the semiconductor. This current-pulse interaction could be the dominant effect

if the energy taken from the pulse for the generation of a new free carrier (via TPA) is less

than the energy which this new carrier gains by being accelerated in the field of the pulse.

This is the principal point which distinguishes our approach from previous work. In fact we

will concentrate entirely on the phenomena associated with the current carried by the free

carriers and neglect other processes including the energy loss due to the production of the

said carriers.

Let us now examine if our model is relevant to any realistic physical situation. The

3

following example clearly demonstrates that the answer is in the affirmative: In order to

generate a free carrier via TPA in InSb we need 0.234 eV. This very carrier, when accelerated

in the pulse field, gains from the pulse an energy E = (eE)2/ (2m,"w2) , where E is the peak

amplitude of the laser electric field, mZ is the effective electron mass, w is the laser frequency.

The energy I is related to the peak intensity and the wavelength of a circularly polarized

laser pulse by the formula €(el/) = 7.02 x 10-'21(W/cm2)X2(pm). With I := 3.3 x 10gW/cm2

at X = lopurn, this formula gives € = 2.32 eV which is almost 10 times greater than the energy

needed (by TPA) to generate one free carrier! It is also clear, that for a sufficiently large

laser intensity, such a situation will pertain for a wide variety of semiconductors.

The preceding discussion suggests that for appropriately chosen systems, we can safely

neglect the TPA contribution to the imaginary part of the effective third-order polarization,

and concentrate only on the fast nonlinear current generated by TPA. Thus, the wave

equation for the optical field of the laser pulse (assuming that all quantities along the

pulse propagation direction, i.e., dong the waveguide) can be written its

(3) d2E ,d2E i3j -- c*- = -47r- at2 8x2 at

Here c, = c/&, c is the speed of light in vacuum, E is the optical dielectric constant of the

medium, and j(x:t) is the pulse induced current primarily carried by the electrons created

by TPA. To complete the model, we need a 'constituent equation' connecting this induced

current with the electric field of the laser pulse.

Let A N ( z , t') = (aiv(z, t')/dt')dt' denote the small group of free carriers generated at

time t'. After being accelerated in the field of the laser pulse, at time t , this group will have

acquired a velocity (starting from rest)

V(Z, t ) = -- e 1,' dt"E(z, t") mZ

Integrating Aj =' -eAN(z*, t')v(s, t), the total induced current will be

(4)

4

Equation ( 5 ) , connecting the pulse driven current of free carriers with the pulse electric field,

forms the essence of our model. This induced nonlinear current is of fifth-order in the field

amplitude. Substituting Eq. ( 5 ) in Eq. (3) we obtain the following wave equation governing

the pulse propagation: d2E ,d2E 4re2 NE - at2 -c*2=-- dX ma E

Equations (2) and (6) form a set of coupled nonlinear equations describing the dynamics

of a short intense laser pulse in a semiconductor when the pulse first generates large densities

of free charge carriers through the resonant TPA process and, then accelerates them to

produce a large nonlinear current. Needless to say that the characteristics of the pulse

will be profoundly changed by this induced current. This current will strongly affect the

dispersion properties of the medium, and will also cause pulse damping.

This damping can be physically understood by analyzing the response of the “newly

born” free carriers to the pulse. Born at zero velocity, these carriers are accelerated in the

pulse field, and thus gain considerable amount of energy from the pulse. Thus TPA, in our

model, plays a twefold role: first it generates a plasma of excess carriers, which changes the

refractive properties of the medium and, second, it contributes to the ‘absorbing’ properties

of the medium through the same term. This absorption (due to the energy expended in .

acceleration) is quite different from the conventional absorption (due to the energy expended

in putting electrons in the conduction band) associated with TPA.

.

Going back to Eq. (Z), and noting that I cy \El2, it seems that the TPA induced free

carrier nonlinearity (proportional to N E) is equivalent to a nonlinearity described by a

7i5’ optical susceptibility. Note that the free carrier absorption is neglected in our model.

Let us make a digression to justify this assumption.

At high carrier densities (2 1017cm-3) scattering time of free carriers due to carrier-

carrier interactions is of the order of to = 100 fs, and scattering times for phonon and ionized

impurities are of the orderbf a few picoseconds. However, in a high intensity radiation field,

the interaction time is modified to t = to(z/50)s/2, where is the t o t d average energy of

5

the carriers (Le. = E + EO), and EO is the average energy in the absence of an external

field. The exponent s = 1 if the optical phonon scattering is dominant, and s = 3 for the

momentum-loss carrier scattering on the ionized impurities or on other carriers [SI. For

problems of interest to this work, the electrons, in the intense radiation field, acquire energy

of the order of 1 eV. For these energies, the shortest relaxation time is of the order of several

picoseconds. Thus our model could only be applied to fast processes which take place on

shorter time scales. Fortunately, subpicosecond times are precisely the times of interest to

us.

In order to gain some insight into the physical processes, let us consider first a model

problem of a square-shaped circularly polarized wave packet

1 E = 2(f + i2)eivEo + c.c (7)

traveling through the semiconductor. Here EO = Eo(z,t) is a real-valued amplitude, and

p = ( ~ ( x , t ) is the phase of the circularly polarized laser pulse.

Introducing the new variables = x - c,t, 7 = t , we can reduce Eqs. (1)-(2) to

where E = EOefp is the complex amplitude.

Consider a right-going square shaped laser pulse with amplitude Eo and length 1 = C,TL,

Eo = const if - I < < < 0 EoK) =

otherwise lo For quasi-stationary propagation of the pulse c,a/a< >> a/&) which changes Eqs. (7)-(8)

to

- E: d N 86

-c*- -

6

Integrating Eq. (12) we obtain the carrier density excited by the pulse,

.V, = const 0 - 0

?Jo - (CY[/C*)E$< -1 < < < 0 (13)

No + (~ i l /c , )E$l < > -1

which coupled with Eq. (ll), leads to the following equation governing the nonlinear evolu-

tion of the phase y (c , t ) inside the pulse,

where up0 = ( 4 ~ e ” - ? J ~ / r n H ~ ) ~ / * is the carrier plasma frequency corresponc,,ig to the carrier

density in front of the pulse i.e., the value of the ambient uniform density before the pulse

enters the semiconductor. The ansatz

leads to the following exact solution of the nonlinear Eq. (14)

where u o and Qo are the initial frequency and phase of the pulse. The complex.amplitude

E. then, becomes

W O

W E = Eo exp(icp) = - eo exp + iRe(+(r))

Equations (16) and (18) display two important consequences of the nonlinear dynamics

of the pulse propagation: 1) Equation (16) indicates that the pulse is upshifted in frequency,

and this upshift is continuous in time, and 2) Equation (18) clearly shows that the pulse

amplitude damps in time.

Even this simplified mbdel shows that the nonlinear self-consistent interaction . - of the

pulse with the plasma of excess-carriers may provide a tunable source of high frequency

radiation. _ .

7

Note that our quasi-stationary approximation is valid when w2 >> i.e., when the

carrier plasma is transparent to the laser pulse. In this case. according to Eq. (16), large

frequency shifts will require large interaction times or, equivalently, the pulse propagation

over large distances inside the semiconductor waveguide. For such large traversal times,

collisions and other complicating effect will render the simple analytical calculation quite

useless. In order to deal with the physics of large frequency shifts which will require W; >> w2.

we must work directly with Eqs. (2) and (6).

In many real situations even when a pulse enters a semiconductor waveguide with initially

underdense carrier plasmas, it can boost the carrier plasma density to near-critical or even

over-critical values by generating excess carriers via the TPA process. In this case a much

"stronger and faster interaction" between the pulse and the generated plasmas takes place.

We carried out a numerical simulation of this process described by the nonlinear set of

equations (2) and (6). The simulation reveals a number of new features of this interaction;

it also provides reasonable arguments for explaining the nature of the spect;rum observed in

experiments.

Notice that we must solve three coupled nonlinear equations: Eq. ( 6 ) 'representing two

full wave equations for the two components of the laser field, and Eq. (2) for the carrier

density. We use a finite difference scheme, second order in both time and space. The results

for the pulse propagation through a waveguide of the well-known 3-5 semiconductor InSb

will be presented below.

The simulation employed an initially Gaussian (for [El) pulse with TFWHM = 200 fs ( f i l l

IVidth at Half Maximum) and wavelength X = 10.6pm. It propagates from left to right in

the InSb waveguide in which the free carrier density prior to the pulse is No = 0.01 iV,

where &VP is the carrier density at w=wp, the carrier plasma frequency. For the pulse tuned

at X = 10.6pmI -Vm = 2.37 x l O " ~ m - ~ .

Although we solve the'full wave equations for Ey and Ez components of the circularly

polarized laser field, we display only the pulse envelope IEI versus z in Fig. 1. The envelope is

8

normalized to the value corresponding to the initial peak intensity IO = 1O9W/cm2. In Fig. 2,

we show the spectral content of the wave at the initial and final times of the simulation.

Figure la reveals that after a rather short period, t = O.OBps, the pulse amplitude is

significantly damped, and the envelope profile has developed strong short scale modulations.

In this as well as later pictures, the dashed line shows the initial location of the pulse. In

Fig. l b (.33ps) we see that the pulse amplitude has damped to less than half its initial value.

At the same time we observe that the pulse has developed a double-peaked structure; the

initial right-going pulse is "split" into two: a transmitted and a reflected pulse. Later parts

of the pulse are reflected from the region where the initial pulse, via TPA. created a dense

enough carrier plasma. In Fig. IC, the trend shown in Fig. l b continues, the reflected and

the transmitted waves have moved further apart, and slow amplitude damping continues.

These figures show that one can distinguish at least two stages in the pulse dynamics.

The first stage lasts up to 0.33~s. At this stage the pulse evolves as a whole and the

dominant process is its amplitude damping. This transient period ends with the formation

of the transmitted and reflected pulses. Both branches have clearly pronounced sharp leading

edges and relatively stretched trailing edges. The dynamics of these pulses shows the strong

GVD effect which, in turn, is enhanced by nonlinear changes in the refractive index. This

dynamics in the space-time domain cannot be completely understood without analysis in

the frequency domain.

The spectral analysis, Fig. 2, shows that both the transmitted and the reflected pulses

are significantly upshifted in frequency. The transmitted pulse is upshifted more than the

reflected one. Its spectrum is represented by a set of almost evenly spaced peaks in the

frequency interval 2v8 u < 2.6vi, where ut = 28.3THz is the initial laser frequency. The

biggest peak is centered at u = 2 . 6 ~ ~ . The spectrum of the reflected pulse is shown by the

dashed line. It consists of a single well pronounced peak centered at 1.8vi.

Similar results are obtgined for several other semiconductors; GaAs, for example.

The prediction of this spectacular frequency upshift (photon acceleration) caused by the

9

strong and fast nonlinear interaction of the semiconductors with ultrashort laser pulses is

the principal result of this paper.

Sluch of the blueshift exhibited in Fig. 2 seems to take place in the first, stage of the pulse

dynamics when the transmitted and the reflected pulses are not yet well separated and are

still strongly influenced by the dramatic and rapid changes in the newly born excess carrier

plasma. The sudden, almost uniform (throughout the width of the short propagating pulse)

and large increase in plasma density induces a sudden and large temporal variation of the

index of refraction of the background medium. The pulse frequency must upshift in order

to satisfy the new dispersion relation defined by the new index of refraction. This simple

and qualitative explanation constitutes the essential physics unravelled in our numerical

modeling. The amount of blueshift is determined by the pulse intensity as well as the

traversal time in the cavity; the system is tunable, i.e., an appropriate choice of parameters

can lead to the desired blueshift. Such frequency blueshifts are also observed in short intense

pulses converting an initially neutral gas into a dense plasma via multiphoton ionization 161.

li’hat does the frequency upshift do to the subsequent pulse dynamics. First of all, it

means that the frequency is no longer tuned near the two-photon resonance, hence the TPA

is automatically “switched off.” Thus the transmitted and the reflected pulses revert back

to the linear mode of propagation. But unlike the initial pulse propagation at early stages

when it enters the waveguide, the transmitted and the reflected pulses undergo enhanced

influence of the positive GVD effect; this influences the final shapes of the pulses according

to their spectral content.

LVe would like to emphasize that our numerical modeling employs the full wave equation

instead of the reduced envelope equation. The simulation results clearly show that the en-

velope equations based on SVEA fail to adequately describe not only the fast processes that

occur on short time scales of a wave period but also processes peculiar to a strongly inho-

mogeneous medium. For iristance, one cannot describe the generation and 8urt;her dynamics

of the backward going reflected waves within the framework of the SVEA, i.e., Eq. (1).

10

For narrow gap semiconductors like InSb. the laser wavelength, for w ich TPA is domi-

nant, should be in the range 7.3pm< X < 14.6pm. At these wavelengths the generation of

femtosecond laser pulses pushes the laser technology to its current limit (see Seifert et ai.

in Ref. [l]). On the other hand, femtosecond lasers operating at - 1,um wavelengths are

widely accessible. Such lasers can be used for wider gap semiconductor such as GaAs. For

GaAs, however the TPA coefficient is several orders of magnitude smaller than for InSb.

Consequently higher laser intensities will be needed to duplicate the processes described

above.

We have just demonstrated that in a semiconductor a sufficiently intense laser pulse

tuned near a two-photon resonance first generates a large excess of free carriers (via TPA),

and then accelerates them to create a large time varying current. The nonlinear interaction

of the propagating pulse with this self-generated current induces (among other things) a

spectacular blueshift in the pulse frequency; the semiconductor is turned into an efficient

photon accelerator. The harnessing of this process for constructing a tunable radiation

source is an obvious application. The emerging pulse, in addition, may serve as a useful

diagnostic as it carries valuable information about the macroscopic characteristics of the

medium, in interaction with which it was born.

11

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[I] S.A. Akhmanov, V.A. Vysloukh. and A S . Chirkin, Ci . 'LS of Femtosecond Laser Pulses,

(New York, AIP, 1992); F. Seifert et al., Opt. Lett. 19, '2009 (1994); 0. Kittelmann et al.,

Opt. Lett. 21, 1159 (1996).

[2] A. hliller, D.A.B. Miller, and S.D. Smith, Advances in Physics 30, 697 (1981); A. Miller

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Heckenmerg, J. Opt. SOC. Am. B 12, 393 (1995).

[3] P.T. Guerreiro et al. Opt. Lett. 21,.659 (1996); A.S. Rodrigues. .LT. Santagiustina, and

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(1993).

[-I] S.W. Koch, S. Smitt-Rink, and H. Haug, Phys. Stat. Sol. (B) 106, 135 (1981); T. El-

saesser, H. Lobentanzer, and W. Kaiser, Appl. Phys. Lett. 47, 1190 (1985); K.W. DeLong

et al.. J. Opt. SOC. Am. €3 6, 1306 (1989); D. Richardson, E.M. Wright, and S.W. Koch,

Phys. Rev. A 41, 1620 (1990); D. Richardson et al., Phys. Rev. A 44, 628 (1991); E.W.

Van Stryland e t al., Opt. Lett. 10, 490 (1985).

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12

FIGURE CAPTIONS

FIG. 1. Plots of \El = (E: +Ez)1’2 versus x at different times for an initially (- - -) Gaussian

pulse with wavelength X = 10.6pm, and width 200fs.

. FIG. 2. Spectral content of the initial (-. --), reflected (- - -) and the transmitted (-) pulses.

E

E

1 .o 0.8

0.6

0.4

0.2

0.0 - 1

r 1 ~~ ~ -

00 - 50 0 50 100 x (microns)

0.0

0.6

0.4

0.2

0.0 - 1 00 -5 0 0 5 0 100

x (microns) 1.0 I I I I I

I I 1. I I \

0.8 }

-150-100 -50 0 5 0 100 150

x (microns)

Fig. 1 (a) - (c)

0 .25

*= 0 . 2 0 v) C Q)

0.15

2 0.10 4- 0 Q) Q 0.05 cr)

0.00 0 0.5 1 1.5 2 2.5 3 3.5

Frequency (x 28.3 THz)

Fig. 2